## Introduction

Advances in technology have led to an increase in large genetic association studies of disease. Along with the ability to look at large numbers of single nucleotide polymorphisms (SNPs) has come the need for improved methods of statistical correction for multiplicity. The Bonferroni procedure is the simplest and most often used method of correction. However, the Bonferroni procedure is well known to be overly conservative in the presence of correlation (Han et al., 2009).

Permutation tests implicitly account for correlation through the use of the data vectors, thereby improving power over Bonferroni-type methods. Permutation tests are also the only multiple comparison procedures capable of exact error control for small or moderate sample sizes. Multiple comparison procedures based on permutations have several other attractive features. One is that they implicitly reduce the penalty for comparisons when the events are rare (Westfall & Troendle, 2008), such as when a SNP is too uncommon to produce a small enough p-value to affect the permutational correction for multiplicity. The permutational procedure will essentially consider that SNP not tested, effectively reducing the correction factor. This is an important advantage when some of the SNPs under study have rare alleles. Another advantage of permutation tests is their ability to handle multiple tests of the same hypothesis easily. It is quite common in genetic association studies for several different modes of inheritance to be considered, typically dominant, recessive, and multiplicative. Each of these modes of inheritance leads to different tests of the null hypothesis. Permutational procedures need only consider the minimum p-value across all tests and SNPs to produce adjusted p-values that account for both the multiple SNPs and multiple tests applied to each SNP.

Another statistical challenge is providing improved methods for family-based studies. Some diseases, like birth defects, are well suited to collection of genetic information on triads (case child, mother, father). The use of triads avoids two problems: (1) ascertainment bias inherent in control selection (Schlesselman, 1982) and (2) population stratification where the case groups may contain different proportions of an ethnic group than the control group. Both of these lead to excess type I errors in tests of association using case–control designs (Lee & Wang, 2008). Triads allow methodology conditioned on the parental genotypes that is robust to population stratification. A very common genetic association test for a single bi-allelic locus (e.g., SNP), based only on triad data, is the transmission disequilibrium test (TDT) (Spielman et al., 1993). This test can be obtained as a likelihood ratio test for the child's genotype in a multiplicative model, conditioned on the parental genotype.

In this report, we describe a one-sample permutational approach for the inheritance-association hypothesis, and show how it can be used when correcting for multiplicity. The resulting multiple comparison procedure permits testing multiple SNPs and multiple tests of each SNP to allow for different inheritance models. We show that the method strongly controls the familywise error rate (FWE), regardless of sample size. Simulations show the permutational procedure to have more power than the Bonferroni procedure under varying inheritance modes, risk allele proportions, and SNP correlations. We analyze a study of candidate genes in neural tube defect (NTD [MIM #182940]) triads as well as a study of oral cleft (OFC1 [MIM #119530]) triads in Ireland, showing that the smallest adjusted p-values from the permutational procedure can be approximately half that of the Bonferroni procedure.